118 XIII INTEGRATION By virtue of the definitions, it is enough to show that if u e £f, v e f and u g v, then ^(u) <; n*(v). Since - u e ^ it follows that v - u = v + (— M) is defined throughout X, belongs to «/ and is §0; hence 0 g p*(v - w) = /i*(t? + ( - M)) = A**(fl) + ju*( - w) by virtue of (13.5.3). We shall often write \ fd\L or | */(#) rfju(x) (resp. | fdfior j f(x instead of /**(/) (resp. ^*(/)). For any subset A of X, we put ju*(A) = /i*(<pA) and ^*(A) = where <^)A is the characteristic function of A (12.7). These numbers /**(A) and ^*(A) are ^0 (possibly H-oo); they are called, respectively, the outer measure and inner measure of A. Example (13.5.11) Let 1 be Lebesgue measure on R and let I = ]#, £[ be an open interval in R. We shall show that /L*(I) == b — a (which is equal to + oo if &=+oo or<2=— oo). If a < a' < V < b, then there exists a continuous mapping/of R into the unit interval [0, 1], with support contained in [<2, £], and equal to 1 throughout [a', 6'] (4.5.2). We have (+c°f(t)dt^bf-af. J — oo Conversely, for each function g e «#"R(R) such that 0 ^ g ^ (pl, we have f + °V(0dt^b-a. Hence it follows that A*(I) = b-a. J — oo Now let U be any 0/?ew sef in R. Then its connected components (3.19) are open intervals ((3.19.1) and (3.19.5)), and the set of connected components is at most denumerable, because each component contains a point of the denumerable set Q, and the components are pairwise disjoint. Hence if they are lk = ~\ak, Afc[, then <?>u = Z Wiv anc^ therefore ((12.7.4) and (13.5.4)) k (13.5.11.1) 6. NEGLIGIBLE FUNCTIONS AND SETS A mapping/: X -» R is said to be negligible (with respect to the measure ju), or it-negligible, if Ai*(|/|) = 0. Then afis also negligible for all a ^ 0 in R; and if \g\ <£ |/|, then g is negligible.exists an> the